Step |
Hyp |
Ref |
Expression |
1 |
|
peano2uz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝐾 ) ) |
2 |
|
eluzelre |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℝ ) |
3 |
|
ltp1 |
⊢ ( 𝑁 ∈ ℝ → 𝑁 < ( 𝑁 + 1 ) ) |
4 |
|
peano2re |
⊢ ( 𝑁 ∈ ℝ → ( 𝑁 + 1 ) ∈ ℝ ) |
5 |
|
ltnle |
⊢ ( ( 𝑁 ∈ ℝ ∧ ( 𝑁 + 1 ) ∈ ℝ ) → ( 𝑁 < ( 𝑁 + 1 ) ↔ ¬ ( 𝑁 + 1 ) ≤ 𝑁 ) ) |
6 |
4 5
|
mpdan |
⊢ ( 𝑁 ∈ ℝ → ( 𝑁 < ( 𝑁 + 1 ) ↔ ¬ ( 𝑁 + 1 ) ≤ 𝑁 ) ) |
7 |
3 6
|
mpbid |
⊢ ( 𝑁 ∈ ℝ → ¬ ( 𝑁 + 1 ) ≤ 𝑁 ) |
8 |
2 7
|
syl |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ¬ ( 𝑁 + 1 ) ≤ 𝑁 ) |
9 |
|
elfzle2 |
⊢ ( ( 𝑁 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝑁 + 1 ) ≤ 𝑁 ) |
10 |
8 9
|
nsyl |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ¬ ( 𝑁 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
11 |
10
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐾 ∈ ℤ ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) → ¬ ( 𝑁 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
12 |
|
nelneq2 |
⊢ ( ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝐾 ) ∧ ¬ ( 𝑁 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → ¬ ( ℤ≥ ‘ 𝐾 ) = ( 𝑀 ... 𝑁 ) ) |
13 |
1 11 12
|
syl2an2 |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐾 ∈ ℤ ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) → ¬ ( ℤ≥ ‘ 𝐾 ) = ( 𝑀 ... 𝑁 ) ) |
14 |
|
eqcom |
⊢ ( ( ℤ≥ ‘ 𝐾 ) = ( 𝑀 ... 𝑁 ) ↔ ( 𝑀 ... 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ) |
15 |
13 14
|
sylnib |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐾 ∈ ℤ ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) → ¬ ( 𝑀 ... 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ) |
16 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
17 |
16
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐾 ∈ ℤ ) ∧ ¬ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
18 |
|
nelneq2 |
⊢ ( ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) ∧ ¬ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) → ¬ ( 𝑀 ... 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ) |
19 |
17 18
|
sylancom |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐾 ∈ ℤ ) ∧ ¬ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) → ¬ ( 𝑀 ... 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ) |
20 |
15 19
|
pm2.61dan |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐾 ∈ ℤ ) → ¬ ( 𝑀 ... 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ) |